Consider any infinite conditional convergent series of the form: $Sum_0=\sum_{n=1}^\infty f(n) + g(n) + h(n) = C_0$. Here each 'term' consists of 3 separate functions $f(n), g(n), h(n)$. Now assume two oerations:
we reorder the functions in each 'term' as follows: $Sum_1=\sum_{n=1}^\infty g(n) + h(n) + f(n) = C_1$
we 're-bracket'/regroup the 'terms' and rewrite the new series as follows: $Sum_2=f(1) + g(1) +\sum_{n=1}^\infty h(n) + f(n+1)+g(n+1) = C_2$
In each of the following case the $Sum$ does not change, thus: $Sum_0=Sum_1=Sum_2=C_0=C_1=C_2.$
Query: Now, if we both reorder the functions in the term as well as regroup them as follows: $Sum_3=g(1) + h(1) +\sum_{n=1}^\infty f(n) + g(n+1)+ h(n+1) = C_2$. Is it always true that the final sum will always remain unchanged i.e. $Sum_0=Sum_3=C_0=C_3$?